33% of adults say cashews are their favorite kind of nut. You randomly select 12 adults and ask each to name their favorite kind of nut. Find the probability that the number who say cashews are their favorite nutis (a)exactly three, (b) at least four, (c) at most two
To find the probability in these scenarios, we'll use the concept of binomial probability. Binomial probability helps us calculate the probability of a certain number of successes in a fixed number of independent trials.
In this case, we have 12 independent trials (randomly selecting 12 adults), where the probability of success (an adult saying cashews are their favorite nut) is 33% or 0.33.
(a) To find the probability of exactly three adults saying cashews are their favorite nut, we'll use the binomial probability formula:
P(X = k) = (n C k) * p^k * (1 - p)^(n - k)
where:
- P(X = k) represents the probability of getting exactly k successes,
- n is the number of trials (12 in this case),
- k is the number of successes (3 in this case),
- p is the probability of success (0.33),
- (n C k) is the combination formula to calculate the number of ways to choose k successes from n trials.
Using these values, we can calculate the probability of exactly three adults saying cashews are their favorite nut:
P(X = 3) = (12 C 3) * 0.33^3 * (1 - 0.33)^(12 - 3)
Calculating this expression:
P(X = 3) = 220 * 0.33^3 * 0.67^9 ≈ 0.2387
Therefore, the probability that exactly three adults say cashews are their favorite nut is approximately 0.2387 (or 23.87%).
(b) The probability of "at least four" adults saying cashews are their favorite nut means finding the probability of having four or more adults with that preference. To calculate this, we need to find the probabilities for four, five, six, ..., up to twelve adults. Then, we sum up these individual probabilities.
P(X ≥ 4) = P(X = 4) + P(X = 5) + ... + P(X = 12)
We can find each probability using the same binomial probability formula as before and sum them up.
P(X ≥ 4) = P(X = 4) + P(X = 5) + ... + P(X = 12)
= (12 C 4) * 0.33^4 * 0.67^8 + (12 C 5) * 0.33^5 * 0.67^7 + ... + (12 C 12) * 0.33^12 * 0.67^0
By calculating this expression, you can find the probability of at least four adults saying cashews are their favorite nut.
(c) The probability of "at most two" adults saying cashews are their favorite nut means finding the probability of having two or fewer adults with that preference. This is equal to the sum of the probabilities of zero, one, and two adults having that preference.
P(X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2)
= (12 C 0) * 0.33^0 * 0.67^12 + (12 C 1) * 0.33^1 * 0.67^11 + (12 C 2) * 0.33^2 * 0.67^10
By calculating this expression, you can find the probability of at most two adults saying cashews are their favorite nut.